Oh brother, what a mess.
But I think I got it to work. It's a problem with the video card I think.
To get it to work:
Run winecfg on the the Steam bottle with pvz goti. (You might need to have a separate bottle just for this game.)
On the config dialog select the Graphics tab
put the Vector Shader Support to None
and uncheck the Allow Pixel Shader box.
Open up steam and launch the game. Try it twice (without closing steam) the first time for some reason failed for me.
Go to options and unselect 3D acceleration. I would also unselect full-screen but that's my personal preference.
The game should play normally now.
Then close the game and steam, launch winecfg again and restore the Vector Shader Support to Hardware and Allow Pixel Shader box back on.
You should be good to go. Let me know.